Progress through quality Education Finite Difference Method

Finite Difference Method

S. NADARAJA PILLAI

School of Mechanical Engineering

SASTRA University

Thanjavur – 613401

Email: nadarajapillai@mech.sastra.edu

@

Progress through quality Education

Outline

Introduction

Finite Difference Method

Discretization Methods

Forward Backward Central Difference

Schemes

Errors

Finite Difference Method (FDM)



Historically, the oldest of the three



Techniques published as early as 1910 by L. F. Richardson



Seminal paper by Courant, Fredrichson and Lewy (1928)

derived stability criteria for explicit time stepping



First ever numerical solution: flow over a circular cylinder

by Thom (1933)



Scientific American article by Harlow and Fromm (1965)

•

First step in obtaining a numerical solution is to discretize

the geometric domain



to define a numerical grid

•

Each node has one unknown and need one algebraic

equation, which is a relation between the variable value at

that node and those at some of the neighboring nodes.

•

The approach is to replace each term of the PDE at the

particular node by a finite-difference approximation.

•

Numbers of equations and unknowns must be equal

•

Taylor Series Expansion: Any continuous differentiable function, in

•

Higher order derivatives are unknown and can be dropped when the distance between grid points is small.

•

By writing Taylor series at different nodes, x_i-1, x_i+1, or both x_i-1 and

Forward-FDS

Backward-FDS

Central-FDS

1st _{order, order of accuracy P}

kest=1

2nd _{order, order of accuracy P}

kest=1

• Numerical solutions can give answers at only discrete points in

the domain, called grid points.

• If the PDEs are totally replaced by a system of algebraic

equations which can be solved for the values of the flow-field

variables at the discrete points only, in this sense, the original

PDEs have been discretized. Moreover, this method of

discretization is called the

method of finite differences

.

• A partial derivative replaced with a suitable algebraic difference quotient is called finite difference. Most finite-difference representations of derivatives are based on Taylor’s series expansion.

• Taylor’s series expansion:

Consider a continuous function of x, namely, f(x), with all derivatives defined at x. Then, the value of f at a location can be estimated from a Taylor series expanded about point x, that is,

• In general, to obtain more accuracy, additional higher-order terms must be included.

x

x+∆

( )

( )

(

)

...

!

1

...

!

3

1

!

2

1

)

(

)

(

₃ 3

3 2

2 2

+

∆

∂

∂

+

+

∆

∂

∂

+

∆

∂

∂

+

∆

∂

∂

+

=

∆

+

n

n n

x

x

f

n

x

x

f

x

x

f

x

x

f

x

f

x

x

f

(1) Forward difference:

(2) Backward difference:

• Truncation error:

The higher-order term neglecting in Eqs. (a), (b), (c) constitute the

truncation error. The general form of Eqs. (d), (e), (f) plus truncated terms can be written as

Forward:

Backward:

Central:

) ( )

( 1 o x

x f f

x

f _{i i}

i + ∆

∆ − =

∂

∂ ₊

) (

)

( 1 o x

x f f

x

f _{i i}

i + ∆

∆ − =

∂

∂ ₋

2 1

1 _{( )}

2 )

( o x

x f f

x

f _{i i}

i + ∆

∆ − =

∂

∂ _{+ −}

• Second derivatives: * Central difference:

If , then (a)+(b) becomes

* Forward difference:

* Backward difference:

x x

x_i = ∆ _i = ∆

∆ ₊₁

2 2

1 1

2 2

) ( )

( 2 )

( o x

x

f f

f x

f _{i i i}

i ∆ + ∆

+ −

= ∂

∂ _{+ −}

) ( )

( 2 )

( ₂ 2 ₂1

2

x o x

f f

f x

f _{i i i}

i ∆ + ∆

+ −

= ∂

∂ _{+ +}

) ( )

( 2 )

( ₂ 1 ₂ 2

2

x o x

f f

f x

f _{i i i}

i ∆ + ∆

+ −

= ∂

∂ _{− −}

• Mixed derivatives:

• In the solution of differential equations with finite differences, a variety of schemes are available for the discretization of derivatives and the solution of the resulting system of algebraic equations.

• In many situations, questions arise regarding the round-off and truncation errors involved in the numerical computations, as well as the consistency, stability and the convergence of the finite difference scheme.

• Round-off errors:computations are rarely made in exact arithmetic. This means that real numbers are represented in “floating point” form and as a result, errors are caused due to the rounding-off of the real numbers. In extreme cases such errors, called “round-off” errors, can accumulate and become a main source of error.

• Truncation error: In finite difference representation of derivative with Taylor’s series expansion, the higher order terms are neglected by

truncating the series and the error caused as a result of such truncation is called the “truncation error”.

• The truncation error identifies the difference between the exact solution of a differential equation and its finite difference solution without round-off error.